---
tag: VDHW982H70
kind: definition
name: Residually small variety
msc: [08B26]
defines: TH2XQGGFWH
uses: [AJNZJD8442, DASF939439, 9CJCXNYNK8]
sources:
  - url: https://en.wikipedia.org/wiki/Residual_property_(mathematics)
---

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
The variety $\mathcal{V}$ is *residually small* if there is a cardinal $\kappa$ such that every [[subdirectly irreducible algebra|DASF939439]] in $\mathcal{V}$ has fewer than $\kappa$ elements.
A variety that is not residually small is *residually large*.

## Remarks

Equivalently, the subdirectly irreducible members of $\mathcal{V}$ form a set up to isomorphism.
The variety is residually finite when $\kappa$ can be chosen with every subdirectly irreducible member finite, and it has finite residual bound when $\kappa$ can be chosen finite.

Example: the variety of abelian groups is residually small.
Its subdirectly irreducible members are the groups $\Z/p^n\Z$ and $\Z(p^\infty)$ for primes $p$ and $n \ge 1$, which are all countable, so $\kappa = \aleph_1$ works.

Non-example: the variety of all groups is residually large.
For an infinite set $X$, the group of even permutations of $X$ that move only finitely many points is [[simple|9CJCXNYNK8]] and has $\abs{X}$ elements.
A nontrivial simple algebra is subdirectly irreducible, so no cardinal $\kappa$ bounds the sizes.